In mathematics, an **indexed family** is a collection of values that are associated with indexes. For example, a *family of real numbers, indexed by the integers* is a collection of real numbers, where each integer is associated with one of the real numbers.

Formally, an indexed family is the same thing as a function. A function with domain *J* and codomain *X* is equivalent to a family of elements of *X* indexed by elements of *J*. The only difference is that indexed families are thought of as collections instead of as functions. A value is considered to be an element of a family whenever it is an element of the image of the family's underlying function.

When a function *f* : *J* → *X* is treated as a family, *J* is called the *index set* of the family, the functional image *f*(*j*) for *j* ∈ *J* is denoted *x*_{j}, and the mapping *f* is denoted {*x*_{j}}_{j∈J} or simply {*x*_{j}}.

Next, if the set *X* is the power set of a set *U*, then the family {*x*_{j}}_{j∈J} is called a **family of sets indexed by J** .

Read more about Indexed Family: Mathematical Statement, Functions, Sets and Families, Examples, Operations On Families, Subfamily, Usage in Category Theory

### Other articles related to "indexed family, indexed":

**Indexed Family**- Usage in Category Theory

... A diagram is a functor giving rise to an

**indexed family**of objects in a category C,

**indexed**by another category J, and related by morphisms depending on two indices ...

... Consider the

**indexed family**of sets whose index set is the set of natural numbers m, defined as follows Obiviously and therefore there is a natural number m0 such that putting A0,m0=A0 the following relation ...

### Famous quotes containing the word family:

“The value of a *family* is that it cushions and protects while the individual is learning ways of coping. And a supportive social system provides the same kind of cushioning for the *family* as a whole.”

—Michael W. Yogman, and T. Berry Brazelton (20th century)